Each global data object is described by an associated description
vector. This vector stores the information required to establish
the mapping between an object element and its corresponding process
and memory location.

Let A be a generic term for any 2D block cyclicly distributed array.
Such a global array has an associated description vector DESCA.
In the following comments, the character _ should be read as
"of the global array".

NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
array A.
N_A (global) DESCA( N_ ) The number of columns in the global
array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
first column of the array A is
distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix,
and assume that its process grid has dimension p x q.
LOCr( K ) denotes the number of elements of K that a process
would receive if K were distributed over the p processes of its
process column.
Similarly, LOCc( K ) denotes the number of elements of K that a
process would receive if K were distributed over the q processes of
its process row.
The values of LOCr() and LOCc() may be determined via a call to the
ScaLAPACK tool function, NUMROC:

ARGUMENTS

The number of rows to be operated on i.e the number of
rows of the distributed submatrix sub( A ). M >= 0.

P (global input) INTEGER

The number of rows to be operated on i.e the number of
rows of the distributed submatrix sub( B ). P >= 0.

N (global input) INTEGER

The number of columns to be operated on i.e the number of
columns of the distributed submatrices sub( A ) and sub( B ).
N >= 0.

A (local input/local output) COMPLEX*16 pointer into the

local memory to an array of dimension (LLD_A, LOCc(JA+N-1)).
On entry, the local pieces of the M-by-N distributed matrix
sub( A ) which is to be factored. On exit, if M <= N, the
upper triangle of A( IA:IA+M-1, JA+N-M:JA+N-1 ) contains the
M by M upper triangular matrix R; if M >= N, the elements on
and above the (M-N)-th subdiagonal contain the M by N upper
trapezoidal matrix R; the remaining elements, with the array
TAUA, represent the unitary matrix Q as a product of
elementary reflectors (see Further Details).
IA (global input) INTEGER
The row index in the global array A indicating the first
row of sub( A ).

JA (global input) INTEGER

The column index in the global array A indicating the
first column of sub( A ).

DESCA (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix A.

TAUA (local output) COMPLEX*16, array, dimension LOCr(IA+M-1)

This array contains the scalar factors of the elementary
reflectors which represent the unitary matrix Q. TAUA is
tied to the distributed matrix A (see Further Details).
B (local input/local output) COMPLEX*16 pointer into the
local memory to an array of dimension (LLD_B, LOCc(JB+N-1)).
On entry, the local pieces of the P-by-N distributed matrix
sub( B ) which is to be factored. On exit, the elements on
and above the diagonal of sub( B ) contain the min(P,N) by N
upper trapezoidal matrix T (T is upper triangular if P >= N);
the elements below the diagonal, with the array TAUB,
represent the unitary matrix Z as a product of elementary
reflectors (see Further Details).
IB (global input) INTEGER
The row index in the global array B indicating the first
row of sub( B ).

JB (global input) INTEGER

The column index in the global array B indicating the
first column of sub( B ).

DESCB (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix B.

TAUB (local output) COMPLEX*16, array, dimension

LOCc(JB+MIN(P,N)-1). This array contains the scalar factors
TAUB of the elementary reflectors which represent the unitary
matrix Z. TAUB is tied to the distributed matrix B (see
Further Details).
WORK (local workspace/local output) COMPLEX*16 array,
dimension (LWORK)
On exit, WORK(1) returns the minimal and optimal LWORK.

and NUMROC, INDXG2P are ScaLAPACK tool functions;
MYROW, MYCOL, NPROW and NPCOL can be determined by calling
the subroutine BLACS_GRIDINFO.

If LWORK = -1, then LWORK is global input and a workspace
query is assumed; the routine only calculates the minimum
and optimal size for all work arrays. Each of these
values is returned in the first entry of the corresponding
work array, and no error message is issued by PXERBLA.

INFO (global output) INTEGER

= 0: successful exit
< 0: If the i-th argument is an array and the j-entry had
an illegal value, then INFO = -(i*100+j), if the i-th
argument is a scalar and had an illegal value, then
INFO = -i.

FURTHER DETAILS

The matrix Q is represented as a product of elementary reflectors

Q = H(ia)' H(ia+1)' . . . H(ia+k-1)', where k = min(m,n).

Each H(i) has the form

H(i) = I - taua * v * v'

where taua is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
exit in A(ia+m-k+i-1,ja:ja+n-k+i-2), and taua in TAUA(ia+m-k+i-1).
To form Q explicitly, use ScaLAPACK subroutine PZUNGRQ.
To use Q to update another matrix, use ScaLAPACK subroutine PZUNMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(jb) H(jb+1) . . . H(jb+k-1), where k = min(p,n).

Each H(i) has the form

H(i) = I - taub * v * v'

where taub is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in
B(ib+i:ib+p-1,jb+i-1), and taub in TAUB(jb+i-1).
To form Z explicitly, use ScaLAPACK subroutine PZUNGQR.
To use Z to update another matrix, use ScaLAPACK subroutine PZUNMQR.

Alignment requirements
======================

The distributed submatrices sub( A ) and sub( B ) must verify some
alignment properties, namely the following expression should be true: